The general equations of an
n-state Markov model can be expressed in terms of matrices as MP = dP/dt
where P is the column vector P = [P1 P2
... Pn]T and M is the n´n transition matrix with the components

The symbol li,j
signifies the exponential transition rate from state i to state j. This
includes both failure rates and repair rates, and it allows for transitions
in both directions between every pair of states.

For each j, let Cj
denote the jth column vector of M, excluding the diagonal term Mj,j.
Hence Cj has dimension n-1. This vector
represents the transition rates from state j to each of the other n-1 states. We also define (dPj/dt)+ as the rate
of probability flow into state j.

Proposition: At steady-state conditions the quantity (dPj/dt)+/(1-Pj) for any state j depends on the direction of Cj
but is independent of the magnitude.

Proof: Let m denote the (n-1)´(n-1) matrix formed by deleting the jth row and column from the full
transition matrix M, and let p denote the column vector formed
by deleting the jth element from full state vector P. Then the
equations of state give

Setting dp/dt = 0
for the steady-state condition and solving this equation for p gives

Now, letting I
denote the identity row vector I = [1 1 ... 1] of dimension n-1, the
conservation equation can be written as 1 - Pj = Ip.
Hence, multiplying both sides of the above equation by I and
substituting from the conservation equation, we have

We also note that for the
steady-state condition (dPj/dt)+ is equal to the rate
of probability flowing out of state j, which is simply PjICj. Consequently we have

The column vector Cj
can be written as the product of a scalar f and a unit vector
Uj pointing in the direction of Cj.
Making this substitution, the scalar f cancels out, and
we are left with

Hence the left hand
quantity depends on the direction of Cj, but not on the
magnitude, which was to be shown.

Discussion: This proposition tells us that the steady-state
"hazard rate" (the left hand quantity in the preceding equation)
for any state j is independent of the overall rate of outflow from that
state, but it does depend on how that outflow is distributed to the other
states of the model.

Proposition: If the only non-zero component of Cj
is lj,k,
then for steady-state conditions the quantity (dPj/dt)+/(1-Pj) is equal to the reciprocal of the mean time of
transition from state k to state j.

Proof: The reciprocal of the hazard rate is

which is a scalar equal to
the sum of the numbers in the kth column of m-1. We wish
to show that this equals the mean time of transition from state k to state
j. To find an explicit expression for this mean time, it is most convenient
to suppress the "return path" by setting lj,k = 0,
and then begin with the initial condition Pk(0) = 1 and integrate
the time from t = 0 to infinity, weighted according to dPj(t)/dt,
which is the probability density function for entry into state j. Hence the
mean time is given by

Notice that setting lj,k
equal to zero prior to solving for Pj(t) does not affect the time
required for an object to transition from state k to state j, because this
transition does not involve the return path from j back to k. Now, since we
have suppressed the return path, the system equations are simply dp/dt
= mp , which has the explicit dynamic solution

Notice that, since our
initial condition is Pk(0) = 1, and state k is the exclusive
return state from state j, the vector p(0) equals the vector Uj
defined previously. We also have the conservation equation

Substituting into the mean
time integral gives

which was to be shown.

Discussion: In retrospect the proof of this proposition is
slightly superfluous, because it's essentially an immediate corollary of the
previous proposition. Notice that the hazard rate of any state j is
independent of the magnitude of the outflow rate from that state (by the
previous proposition), so we can consider the limit as lj,k
approaches infinity. In this limit the value of Pj approaches
zero, so the hazard rate goes to (dPj/dt)+, which is
just the flow rate into state j (which also equals the flow rate out of state
j). Also, it is stipulated that all the flow from state j goes to state k,
so each element of the steady-state flow proceeds from state k to state j (by
some route) and then immediately jumps back to state k to begin again. The
steady-state flow rate (dPj/dt)+ in the limit as lj,k
approaches infinity is therefore the reciprocal of the mean time of
transition from state k to state j. It follows from the previous proposition
that, for any value of lj,k, the
hazard rate (dPj/dt)+/(1-Pj) equals the reciprocal of
the mean time to transition from state k to state j, which was to be shown.